Problem 34 - Algebra 2

Question

Find each value. Write angle measures in radians. Round to the nearest hundredth. \(\sin \left(2 \cos ^{-1} \frac{3}{5}\right)\)

Step-by-Step Solution

Answer

The value of \( \sin(2 \cos^{-1} \frac{3}{5}) \) is approximately 0.96.

Step 1: Find the cosine angle

First, we need to determine the angle where \( \cos^{-1} \left( \frac{3}{5} \right) \). Let \( \theta = \cos^{-1} \left( \frac{3}{5} \right) \). Therefore, \( \cos \theta = \frac{3}{5} \). The angle \( \theta \) is in the range \([0, \pi]\) for \( \cos^{-1} \).

Step 2: Establish the sine and cosine relationships

Since we have \( \cos \theta = \frac{3}{5} \), using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \), solve for \( \sin \theta \):\( \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left( \frac{3}{5} \right)^2 = \frac{16}{25} \) \( \sin \theta = \sqrt{\frac{16}{25}} = \frac{4}{5} \) (since \( \theta \) is in \( [0, \pi] \), \( \sin \theta \) is positive.)

Step 3: Use double angle formula for sine

Apply the double angle formula for sine: \( \sin(2 \theta) = 2 \sin \theta \cos \theta \). Substitute the values obtained:\( \sin(2 \theta) = 2 \times \frac{4}{5} \times \frac{3}{5} = \frac{24}{25} \).

Step 4: Round the result

The calculated value \( \frac{24}{25} \) is already in a simple fraction, which approximates to 0.96 when rounded to the nearest hundredth.

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions help us find the angle measurements from known trigonometric ratios. Inverse cosine, denoted as \( \cos^{-1} \), is used to determine the angle whose cosine value is a given number. For example, when presented with \( \cos^{-1} \left( \frac{3}{5} \right) \), our task is to find this specific angle, \( \theta \), that exists in the range \([0, \pi]\) radians. This range is crucial because it defines the primary range for the inverse cosine function, ensuring the uniqueness of solutions. Understanding how to use inverse trigonometric functions allows us to unravel angles in diverse mathematical and practical scenarios.

Double Angle Formula

The double angle formula is a powerful tool in trigonometry for expressing trigonometric functions of angles that are twice another angle. Specifically, it simplifies calculations involving double angles. For sine, the double angle formula is given as:

• \( \sin(2\theta) = 2 \sin \theta \cos \theta \)

This formula is essential in problems involving expressions like \( \sin(2 \cos^{-1}(x)) \). After determining the values of \( \sin \theta \) and \( \cos \theta \), using the double angle formula allows us to compute \( \sin(2\theta) \). Thus, we efficiently derive the trigonometric values for double angles with just simple multiplication and known trigonometric values.

Radians

Radians are a way of measuring angles based on the radius of a circle. Instead of dividing a circle into 360 degrees, radians relate the angle to the circumference of the circle directly. One full circle is \( 2\pi \) radians, making conversion between degrees and radians straightforward:

• \( 180 \) degrees = \( \pi \) radians
• \( 1 \) radian is approximately \( 57.2958 \) degrees

Angles can therefore be expressed in this natural mathematical language. Because trigonometric functions are periodic with \( 2\pi \), using radians allows for simple and elegant representations of periodic properties of trigonometric functions. This unit of measure simplifies many calculations and formulas in calculus and other advanced fields.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry. It states that for any angle \( \theta \):

• \( \cos^2 \theta + \sin^2 \theta = 1 \)

This identity is derived from the Pythagorean theorem, relating the sides of a right triangle. When solving for unknown trigonometric values, this equation provides a reliable route. If \( \cos \theta \) is known, \( \sin \theta \) can be calculated, and vice versa, by rearrangement:

• \( \sin^2 \theta = 1 - \cos^2 \theta \)
• \( \cos^2 \theta = 1 - \sin^2 \theta \)

Such identities simplify equations, facilitate solving problems in trigonometry, and aid in verifying the correctness of derived results. In exercises involving angles, this identity is indispensable.